Variational Estimate for the Helium Atom

The Helium Atom is the classic first serious test of the variational method. The exact two-electron problem is not separable because of electron-electron repulsion, but a simple screened-charge trial wavefunction already gives a surprisingly good ground-state energy. The atomic page owns the spectrum, singlet–triplet structure, precision hierarchy, and benchmark interpretation; this page owns the analytic effective-charge calculation.

The general logic of admissible trial sets, optimization, and one-sided energy bounds is summarized in Variational and Bound Methods.

This page uses atomic units:

$$\hbar=m_e=e=4\pi\epsilon_0=1.$$

Energies are in Hartree. For helium, $Z=2$ and

$$H = - \frac12\nabla_1^2 - \frac12\nabla_2^2 - \frac{Z}{r_1} - \frac{Z}{r_2} + \frac{1}{r_{12}},$$

where

$$r_{12}=|\mathbf r_1-\mathbf r_2|.$$

Independent-Electron Trial State

Use a product of two normalized hydrogenic $1s$ orbitals with an adjustable effective charge $\zeta$:

$$\phi_\zeta(r) = \left(\frac{\zeta^3}{\pi}\right)^{1/2} e^{-\zeta r}.$$

The spatial trial state is

$$\Psi_\zeta(\mathbf r_1,\mathbf r_2) = \phi_\zeta(r_1)\phi_\zeta(r_2).$$

For the physical helium ground state, this spatial state is paired with an antisymmetric spin singlet so that the total two-electron state is antisymmetric, as required by the Pauli exclusion principle. The energy expectation below concerns only the spatial Hamiltonian.

Effective Nuclear Charge

The parameter $\zeta$ is interpreted as an effective nuclear charge. If electron-electron repulsion were ignored, each electron would see the full charge $Z=2$. Repulsion screens the nucleus, so one expects

$$1<\zeta<2.$$

For one electron in $\phi_\zeta$,

$$\left\langle - \frac12\nabla^2 \right\rangle = \frac{\zeta^2}{2}, \qquad \left\langle - \frac{Z}{r} \right\rangle = -Z\zeta.$$

For two independent electrons, the one-body contribution is

$$E_{\mathrm{one-body}}(\zeta) = \zeta^2-2Z\zeta.$$

For helium, this is

$$E_{\mathrm{one-body}}(\zeta) = \zeta^2-4\zeta.$$

Electron-Electron Repulsion

The repulsion expectation in this trial state is the standard integral

$$\left\langle \frac{1}{r_{12}} \right\rangle = \int d^3r_1d^3r_2\, \frac{ |\phi_\zeta(r_1)|^2 |\phi_\zeta(r_2)|^2 }{ |\mathbf r_1-\mathbf r_2| }.$$

For two identical $1s$ exponential orbitals,

$$\left\langle \frac{1}{r_{12}} \right\rangle = \frac{5}{8}\zeta.$$

The variational energy is therefore

$$E(\zeta) = \zeta^2 -2Z\zeta + \frac58\zeta.$$

For helium, $Z=2$:

$$E(\zeta) = \zeta^2 - \frac{27}{8}\zeta.$$

Optimized Estimate

Minimize:

$$\frac{dE}{d\zeta} = 2\zeta-\frac{27}{8} = 0.$$

Thus

$$\zeta_\star = \frac{27}{16} = 1.6875.$$

The optimized energy is

$$E_{\mathrm{var}} = - \frac{729}{256} = -2.84765625 \ \text{Hartree}.$$

This is an upper bound on the exact nonrelativistic ground-state energy. It is much better than using unscreened hydrogenic orbitals with $\zeta=2$, which gives

$$E(2)=-2.75 \ \text{Hartree}.$$

The improvement comes from allowing the electrons to screen each other on average.

What the Ansatz Captures

The screened-charge trial state captures:

• nuclear attraction and kinetic-energy balance;
• average screening of each electron by the other;
• the correct spherical symmetry of the ground state;
• the spin singlet structure through a symmetric spatial state.

It misses:

• explicit electron-electron correlation;
• angular correlation between electron positions;
• the electron-electron cusp condition;
• configuration mixing with excited orbitals;
• relativistic, finite-nuclear-mass, and QED corrections.

The missing correlation is why more sophisticated variational wavefunctions include factors depending explicitly on $r_{12}$.

Why This Example Matters

The helium estimate is simple, but it demonstrates a central lesson: a physically meaningful parameter can capture a large part of an interaction effect. The parameter $\zeta$ is not arbitrary curve-fitting; it represents screening, a real physical response of the electronic cloud.

The stationarity, curvature, and scaling logic behind optimizing such nonlinear coordinates is treated in Variational Parameters.

This logic scales into quantum chemistry, where trial spaces become Slater determinants, configuration-interaction expansions, coupled-cluster amplitudes, and correlated basis functions.

Common Mistakes

• Forgetting the spin part of the two-electron state when discussing antisymmetry.
• Interpreting $\zeta$ as the true nuclear charge rather than an effective variational parameter.
• Treating the variational estimate as exact because it is close.
• Comparing to experimental helium energy without accounting for effects outside the nonrelativistic Hamiltonian.
• Missing the electron-electron repulsion term, which is the whole reason the problem is not two independent hydrogenic atoms.

Exercises

1. Minimize $E(\zeta)=\zeta^2-(27/8)\zeta$.

Solution

Set

$$\frac{dE}{d\zeta} = 2\zeta-\frac{27}{8} = 0.$$

Then

$$\zeta_\star=\frac{27}{16}.$$

The minimum energy is

$$E(\zeta_\star) = \left(\frac{27}{16}\right)^2 - \frac{27}{8} \left(\frac{27}{16}\right) = - \frac{729}{256}.$$

2. Show that the optimized effective charge is less than the true nuclear charge for helium.

Solution

For helium, $Z=2$. The optimized value is

$$\zeta_\star=\frac{27}{16}=1.6875.$$

Therefore $\zeta_\star<2$. This reflects screening: each electron partially reduces the nuclear charge felt by the other in the independent-electron approximation.

3. Why does the product trial state miss electron correlation?

Solution

The product state has probability density

$$|\Psi_\zeta(\mathbf r_1,\mathbf r_2)|^2 = |\phi_\zeta(r_1)|^2|\phi_\zeta(r_2)|^2.$$

It treats the two electron positions as statistically independent apart from the shared parameter $\zeta$. It does not explicitly reduce the probability of finding the electrons close together as a function of $r_{12}$. A correlated trial state would include dependence on $r_{12}$.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover, 1996.